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Theorem rmo5 2692
Description: Restricted "at most one" in term of uniqueness. (Contributed by NM, 16-Jun-2017.)
Assertion
Ref Expression
rmo5 (∃*𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑))

Proof of Theorem rmo5
StepHypRef Expression
1 df-mo 2030 . 2 (∃*𝑥(𝑥𝐴𝜑) ↔ (∃𝑥(𝑥𝐴𝜑) → ∃!𝑥(𝑥𝐴𝜑)))
2 df-rmo 2463 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
3 df-rex 2461 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
4 df-reu 2462 . . 3 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
53, 4imbi12i 239 . 2 ((∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑) ↔ (∃𝑥(𝑥𝐴𝜑) → ∃!𝑥(𝑥𝐴𝜑)))
61, 2, 53bitr4i 212 1 (∃*𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wex 1492  ∃!weu 2026  ∃*wmo 2027  wcel 2148  wrex 2456  ∃!wreu 2457  ∃*wrmo 2458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108
This theorem depends on definitions:  df-bi 117  df-mo 2030  df-rex 2461  df-reu 2462  df-rmo 2463
This theorem is referenced by:  nrexrmo  2693  cbvrmo  2702  bdrmo  14544
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